We study properties of positive functions satisfying (E) −∆u+m|∇u| q − u p = 0 is a domain Ω or in R N + when p > 1 and 1 < q < 2. We give sufficient conditions for the existence of a solution to (E) with a nonnegative measure µ as boundary data, and these conditions are expressed in terms of Bessel capacities on the boundary. We also study removable boundary singularities and solutions with an isolated singularity on ∂Ω. The different results depends on two critical exponents for p = p c := N +1 N −1 and for q = q c := N +1 N and on the position of q with respect to 2p p+1. Contents